Strong Types for Direct Logic Carl Hewitt http://plus.google.com/+CarlHewitt-StandardIoT This article is dedicated to Alonzo Church, Richard Dedekind, Stanisław Jaśkowski, Bertrand Russell, Ludwig Wittgenstein. and Ernst Zermelo. Abstract This article follows on the introductory article “Direct Logic for Intelligent Applications” [Hewitt 2017a]. Strong Types enable new mathematical theorems to be proved including the Formal Consistency of Mathematics. Also, Strong Types are extremely important in Direct Logic because they block all known paradoxes[Cantini and Bruni 2017]. Blocking known paradoxes makes Direct Logic safer for use in Intelligent Applications by preventing security holes. Inconsistency Robustness is performance of information systems with pervasively inconsistent information.1 Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over the centuries. In the Inconsistency Robustness paradigm, deriving contradictions has been a progressive development and not “game stoppers.” Contradictions can be helpful instead of being something to be “swept under the rug” by denying their existence, which has been repeatedly attempted by authoritarian theoreticians (beginning with some Pythagoreans). Such denial has delayed mathematical development. This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis for foundations.2 Mathematics here means the common foundation of all classical mathematical theories from Euclid to the mathematics used to prove Fermat's Last [McLarty 2010]. Direct Logic provides categorical axiomatizations of the Natural Numbers, Real Numbers, Ordinal Numbers, Set Theory, and the Lambda Calculus meaning that up a unique isomorphism there is only one model that satisfies the respective axioms. Good evidence for the consistency Classical Direct Logic derives from how it blocks the known paradoxes of classical mathematics. Humans have spent millennia devising paradoxes for classical mathematics. 1 Having a powerful system like Direct Logic is important in computer science because computers must be able to formalize all logical inferences (including inferences about their own inference processes) without requiring recourse to human intervention. Any inconsistency in Classical Direct Logic would be a potential security hole because it could be used to cause computer systems to adopt invalid conclusions. After [Church 1934], logicians faced the following dilemma: 1st order theories cannot be powerful lest they fall into inconsistency because of Church’s Paradox. 2nd order theories contravene the philosophical doctrine that theorems must be computationally enumerable. The above issues can be addressed by requiring Mathematics to be strongly typed using so that: Mathematics self proves that it is “open” in the sense that theorems are not computationally enumerable.3 Mathematics self proves that it is formally consistent.4 Strong mathematical theories for Natural Numbers, Ordinals, Set Theory, the Lambda Calculus, Actors, etc. are inferentially decidable, meaning that every true proposition is provable and every proposition is either provable or disprovable. Furthermore, theorems of these theories are not enumerable by a provably total procedure. Mathematical Foundation for Computer Science Computer Science brought different concerns and a new perspective to mathematical foundations including the following requirements:5 [Arabic numeral superscripts refer to endnotes at the end of this article] provide powerful inference machinery so that arguments (proofs) can be short and understandable and all logical inferences can be formalized establish standard foundations so people can join forces and develop common techniques and technology incorporate axioms thought to be consistent by the overwhelming consensus of working professional mathematicians, e.g., natural numbers [Dedekind 1888], Actors, real numbers [Dedekind 1888], ordinals, sets, lambda calculus, etc. facilitate inferences about the mathematical foundations used by computer systems. 2 Sociology of Foundations “Faced with the choice between changing one’s mind and proving that there is no need to do so, almost everyone gets busy on the proof.” John Kenneth Galbraith [1971 pg. 50] “Max Planck, surveying his own career in his Scientific Autobiography [Planck 1949], sadly remarked that ‘a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.’ ” [Kuhn 1962] The inherently social nature of the processes by which principles and propositions in logic are produced, disseminated, and established is illustrated by the following issues with examples:6 The formal presentation of a demonstration (proof) has not led automatically to consensus. Formal presentation in print and at several different professional meetings of the extraordinarily simple proof in this paper have not lead automatically to consensus about the theorem that “Mathematics proves that it is formally consistent”. New results can sound crazy to those steeped in conventional thinking. Paradigm shifts often happen because conventional thought is making assumptions taken as dogma. As computer science continues to advance, such assumptions can get in the way and have to be discarded. There has been an absence of universally recognized central logical principles. Disputes over the validity of the Principle of Excluded Middle led to the development of Intuitionistic Logic. There are many ways of doing logic. One view of logic is that it is about truth; another view is that it is about argumentation (i.e. proofs).7 Argumentation and propositions have be variously (re-)connected and both have been re-used. Church's paradox [Church 1934] is that assuming theorems of mathematics are computationally enumerable leads to contradiction. In this article, Church’s Paradox is transformed into the fundamental principle that “Mathematics is Open” (i.e. it is a theorem of mathematics that the proofs of mathematics are not computationally enumerable).i i See discussion in this article. 3 New technological developments have cast doubts on traditional logical principles. The pervasive inconsistency of modern large-scale information systems has cast doubt on some logical principles, e.g., Excluded Middle.8 That there are proofs that cannot be expressed through text alone, overturns a long-held philosophical dogma about mathematical theories, i.e., that all theorems of a theory can be computationally generated by starting with axioms and mechanically applying rules of inference Political actions have been taken against views differing from the establishment theoreticians. According to [Kline 1990, p. 32], Hippasus was literally thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” Fearing that he was dying and the influence that Brouwer might have after his death, Hilbert fired9 Brouwer as an associate editor of Mathematische Annalen because of “incompatibility of our views on fundamental matters”10 e.g., Hilbert ridiculed Brouwer for challenging the validity of the Principle of Excluded Middle. [Gödel 1931] results were for Principia Mathematica as the foundation for the mathematics of its time including the categorical axiomatization of the natural numbers. In face of Wittgenstein's devastating criticism, Gödel insinuated11 that he was crazy and retreated to relational 1st order theory in an attempt to salvage his results. Since theoreticians found it difficult to prove anything significant about practical mathematical theories, they cut them down to unrealistic relational 1st order theories where results could be proved (e.g. compactness) that did not hold for practical mathematical theories. In the famous words of Upton Sinclair: “It is difficult to get a man to understand something, when his salary depends on his not understanding it.” Some theoreticians have ridiculed dissenting views and attempted to limit their distribution by political means.12 4 Foundations with strong parameterized types “Everyone is free to elaborate [their] own foundations. All that is required of [a] Foundation of Mathematics is that its discussion embody absolute rigor, transparency, philosophical coherence, and addresses fundamental methodological issues.”13 “The aims of logic should be the creation of “a unified conceptual apparatus which would supply a common basis for the whole of human knowledge.” [Tarski 1940] Note: types in Direct Logic are much stronger than constructive types with constructive logic because Classical Direct Logic has all of the power of Classical Mathematics. Booleans are Propositions although Propositions are not reducible to Booleans: True:Boolean False:Boolean Boolean⊑Proposition1 //each Boolean is a Proposition Boolean≠Proposition1 //some Propositions are not Booleans (3=3) ≠ True //the proposition 3=3 is not equal to True (3=3) ≠ (4=4) //the proposition 3=3 is not equal to the proposition 4=4 (3=4) ≠ False //the proposition 3=4 is not equal to False In Direct Logic, unrestricted recursion is allowed in programs. For example, 14 There are uncountably many Actors. For example, Real∎[